Dynare forums

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var pai, c, n, r, a;
varexo u;
parameters beta, rho, epsilon, omega, phi, gamma;

beta=0.99;
gamma=3;
omega=17;
epsilon=8;
phi=1;
rho=0.95;
model;
a = rho*a(-1)+u;
1/c = beta*r/(c(+1)*pai(+1));
pai*(pai-1)/c = beta*pai(+1)*(pai(+1)-1)/c(+1)+epsilon*phi*n^(gamma+1)/omega-exp(a)*n*(epsilon-1)/(omega*c);
exp(a)*n = c+(omega/2)*(pai-1)^2;
end;
initval;
r=1;
end;

steady_state_model;
a = 0;
pai = beta*r;
c = find_c(0.96,pai,beta,epsilon,phi,gamma,omega);
n = c+(omega/2)*(pai-1)^2;
end;
shocks;
var u; stderr 0.008;
end;

planner_objective(ln(c)-phi*((n^(1+gamma))/(1+gamma)));
ramsey_policy(planner_discount=0.99,order=1,instruments=(r));

function c = find_c(c0,pai,beta,epsilon,phi,gamma,omega)
c = csolve(@nk_ss,c0,[],1e-8,100,pai,beta,epsilon,phi,gamma,omega);
function r = nk_ss(c,pai,beta,epsilon,phi,gamma,omega)
r = pai*(pai-1)/c - beta*pai*(pai-1)/c-epsilon*phi*(c+(omega/2)*(pai-1)^2)^(gamma+1)/omega+(c+(omega/2)*(pai-1)^2)*(epsilon-1)/(omega*c);

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c = find_c(0.96,pai,beta,epsilon,phi,gamma,omega);

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function c = find_c(c0,pai,beta,epsilon,phi,gamma,omega)
c = csolve(@nk_ss,c0,[],1e-8,100,pai,beta,epsilon,phi,gamma,omega);
end

function r = nk_ss(c,pai,beta,epsilon,phi,gamma,omega)
r = pai*(pai-1)/c - beta*pai*(pai-1)/c-epsilon*phi*(c+(omega/2)*(pai-1)^2)^(gamma+1)/omega +(c+(omega/2)*(pai-1)^2)*(epsilon-1)/(omega*c);
end


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equation 3: beta*pai(+1)*(pai(+1)-1)/c(+1)+epsilon*phi*n^(gamma+1)/omega -exp(a)*n*(epsilon-1)/(omega*c)

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c=csolve(@nk_ss,c0,[],1e-8,100....


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r = nk_ss(c,pai,beta,epsilon,phi,gamma,omega)


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r = pai*(pai-1)/c - beta*pai*(pai-1)/c-epsilon*phi*(c+(omega/2)*(pai-1)^2)^(gamma+1)/omega +(c+(omega/2)*(pai-1)^2)*(epsilon-1)/(omega*c);


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a = 0;
pai = beta*r;
c = find_c(0.96,pai,beta,epsilon,phi,gamma,omega);
n = c+(omega/2)*(pai-1)^2;

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nk_ss

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function PI_CPI = find_PI_CPI(PI_CPI0, LAMBDA, beta)
PI_CPI = csolve(@nk_ss,PI_CPI0,[],1e-8,100, LAMBDA, beta);
end

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function r = nk_ss(PI_CPI, LAMBDA, beta)
r = (PI_CPI*LAMBDA)/beta;
end

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Warning: Rank deficient, rank = 0, tol = NaN.
> In dyn_ramsey_static_dyn_ramsey_static_1 (line 152)
  In dyn_ramsey_static>@(x)dyn_ramsey_static_1(x,M,options_,oo) (line 40)
  In csolve (line 112)
  In dyn_ramsey_static (line 55)
  In evaluate_steady_state (line 55)
  In resol (line 104)
  In stoch_simul (line 88)
  In ramsey_policy (line 25)
  In FINOP_RAMSEY (line 650)
  In dynare (line 180) 

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Attempted to access ys_(50); index out of bounds because numel(ys_)=40.
Error in FINOP_RAMSEY_steadystate2 (line 44)
ys_(80)=ys_(50)*(-(ys_(18)*1/(ys_(26)*ys_(15))));
Error in evaluate_steady_state_file (line 54)
        [ys,params1,check] = h_steadystate(ys_init, exo_ss, params);
Error in dyn_ramsey_static (line 61)
    [xx,params,check] = evaluate_steady_state_file(ys,exo_ss,M,options_);
Error in evaluate_steady_state (line 55)
        [ys,params] = dyn_ramsey_static(ys_init,M,options,oo);
Error in resol (line 104)
[dr.ys,M.params,info] = evaluate_steady_state(oo.steady_state,M,options,oo,0);
Error in stoch_simul (line 88)
    [oo_.dr,info,M_,options_,oo_] = resol(0,M_,options_,oo_);
Error in ramsey_policy (line 25)
info = stoch_simul(var_list);
Error in FINOP_RAMSEY (line 650)
ramsey_policy(var_list_);
Error in dynare (line 180)
evalin('base',fname)

evaluate_steady_state: The steady state file computation for the Ramsey problem resulted in NaNs.
evaluate_steady_state: The steady state was computed conditional on the following initial instrument values:
r 1.000000
evaluate_steady_state: The problem occured in the following equations:
Equation(s): 1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 38, 40,
evaluate_steady_state: If those initial values are not admissable, change them using an initval-block.

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PHI2=(Y*UC)/(1-pi*beta*(PI_CPI^(varepsilon-1)));
PHI1=((Y*UC)/(1-pi*beta*(PI_CPI^(varepsilon))))*(varepsilon/(varepsilon-1))*MC;


dynare FINOP_RAMSEY

Configuring Dynare ...
[mex] Generalized QZ.
[mex] Sylvester equation solution.
[mex] Kronecker products.
[mex] Sparse kronecker products.
[mex] Local state space iteration (second order).
[mex] Bytecode evaluation.
[mex] k-order perturbation solver.
[mex] k-order solution simulation.
[mex] Quasi Monte-Carlo sequence (Sobol).
[mex] Markov Switching SBVAR.

Using 64-bit preprocessor
Starting Dynare (version 2016-04-16).

evaluate_steady_state: The steady state file does not solve the steady state for the Ramsey problem.
evaluate_steady_state: Conditional on the following instrument values:
r 1.000000
evaluate_steady_state: the following equations have non-zero residuals:
Equation number 3: -2900105745041947900000.000000
Equation number 10: 6117457973423583000000.000000
Equation number 11: -4.324949
Equation number 14: 7.785978
Equation number 16: -2.646347
Equation number 21: 179102810608.442600
Equation number 24: -76677505772.973389
Equation number 29: -1.000000
Equation number 31: -1.000000

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evaluate_steady_state: The steady state file does not solve the steady state for the Ramsey problem.
evaluate_steady_state: Conditional on the following instrument values:
    r     0.010050
evaluate_steady_state: the following equations have non-zero residuals:
    Equation number 16: -1.000000
    Equation number 29: -1.000000

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PI_CPI(+1)=beta*(UC(+1)/UC)*(1/LAMBDA);      //(16)  EULER EQUATION
PI_CPI=CPI/CPI(-1);                          //(29)  CPI INFLATION IDENTITY


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r = (PI_CPI*LAMBDA)/beta;


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PI_CPI=CPI/CPI(-1);                         

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PI_CPI=1

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evaluate_steady_state: The steady state for the Ramsey problem could not be computed.
evaluate_steady_state: The steady state computation stopped with the following instrument values::
    r     0.010050
evaluate_steady_state: The following equations have non-zero residuals:
    Equation number 17: -0.000079
    Equation number 26: 0.000114
    Equation number 29: -0.000171
    Equation number 30: -0.000040

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Error using print_info (line 42)
Blanchard Kahn conditions are not satisfied: no stable equilibrium

Error in stoch_simul (line 94)
    print_info(info, options_.noprint, options_);

Error in ramsey_policy (line 25)
info = stoch_simul(var_list);

Error in FINOP_RAMSEY (line 660)
ramsey_policy(var_list_);

Error in dynare (line 223)
evalin('base',fname) ;

initval r=-log(beta)=0.010050 does not solve the model and gives rise to small residuals

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Warning: Rank deficient, rank = 0, tol = NaN.
> In dyn_ramsey_static_dyn_ramsey_static_1 (line 152)
  In dyn_ramsey_static>@(x)dyn_ramsey_static_1(x,M,options_,oo) (line 43)
  In csolve (line 112)
  In dyn_ramsey_static (line 58)
  In evaluate_steady_state (line 120)
  In resol (line 104)
  In stoch_simul (line 83)
  In ramsey_policy (line 25)
  In GM_2005 (line 542)
  In dynare (line 223)

evaluate_steady_state: The steady state computation for the Ramsey problem resulted in NaNs.
evaluate_steady_state: The steady state computation resulted in the following instrument values:
    i     0.010050
evaluate_steady_state: The problem occured in the following equations:
    Equation(s): 1, 33, 34,
Error using print_info (line 140)
Ramsey: The steady state computation resulted in NaN in the static first order conditions for optimal policy
Error in stoch_simul (line 94)
    print_info(info, options_.noprint, options_);
Error in ramsey_policy (line 25)
info = stoch_simul(var_list);
Error in GM_2008 (line 542)
ramsey_policy(var_list_);
Error in dynare (line 223)
evalin('base',fname) ;

\documentclass[11pt]{article}
\usepackage{amsmath, amssymb}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{float}
\usepackage{natbib}
\bibliographystyle{agsm}
\begin{document}

\title{Running the Ramsey Policy on a Non-Linear\\ Gali \& Monacelli (2005) Model using Dynare}
\author{Justas Dainauskas\footnote{University of York, Department of Economics and Related Studies, Heslington, York, YO10 5DD, United Kingdom. E-Mail: jd1090@york.ac.uk}}
\maketitle

\section{The Model}
Consider the ubiquitous \cite{GaliMonacelli(2005)} model of a small open economy in its non-linear form. The only real difference here is that the capital stock is present in the model and the dynamics of inflation are summarised in a recursive way similar to the \texttt{NK\_baseline.mod} in the dynare example folder. The trade balance is specified in a slightly different way, but it makes no real difference. There are no additional frictions introduced into the model and the notation used is consistent with \cite{GaliMonacelli(2005)} as much as possible, so all of the equilibrium conditions are standard:
\begin{equation}
Y_t=C_t+I_t+NX_t
\end{equation}
\begin{equation}
NX_t=X_t-M_t
\end{equation}
\begin{equation}
X_t=\alpha\left(\frac{P_{H,t}}{E_tP_{F,t}}\right)^{-\eta}C_t^*
\end{equation}
\begin{equation}
M_t=\alpha\left(\frac{E_tP_{F,t}}{P_t}\right)^{-\eta}C_t
\end{equation}
\begin{equation}
C_{H,t}=(1-\alpha)\left(\frac{P_{H,t}}{P_{t}}\right)^{-\eta}C_t
\end{equation}
\begin{equation}
C_{F,t}=\alpha\left(\frac{E_tP_{F,t}}{P_{t}}\right)^{-\eta}C_t
\end{equation}
\begin{equation}
C_t+I_t+\mathbb{E}_t\left[D_{t+1} \mathit{\Lambda}_{t,t+1} \mathit{\Pi}_{t+1}\right]=\frac{D_t}{P_t}+\frac{W_tN_t}{P_t}+\frac{R_tK_{t-1}}{P_t}+\frac{\mathit{\Xi}_t}{P_t}
\end{equation}
\begin{equation}
\mathit{\Xi}_t=P_{H,t}Y_t-W_tN_t-R_tK_{t-1}
\end{equation}
\begin{equation}
\mathbb{E}_t\left[\mathit{\Lambda}_{t,t+1}\right]=\exp(-i_t)
\end{equation}
\begin{equation}
\mathbb{E}_t\left[\mathit{\Pi}_{t+1}\right]=\mathbb{E}_t\left[\frac{P_{t+1}}{P_t}\right]
\end{equation}
\begin{equation}
\frac{W_t}{P_t}=\frac{N^\varphi}{\mathbb{U}_{c,t}}
\end{equation}
\begin{equation}
\mathbb{U}_{c,t}=C^{-\sigma}
\end{equation}
\begin{equation}
\mathbb{E}_t\left[\mathit{\Lambda}_{t,t+1}\mathit{\Pi}_{t+1}\right]=\beta\left[\frac{\mathbb{U}_{c,t+1}}{\mathbb{U}_{c,t}}\right]
\end{equation}
\begin{equation}
\frac{R_{t+1}}{P_{t+1}}=\mathbb{E}_t\left[\left(\frac{\mathbb{U}_{c,t+1}}{\mathbb{U}_{c,t}}\right)\left(\frac{1}{\beta}-(1-\delta)\right)\right]
\end{equation}
\begin{equation}
Y=\frac{A_tK_{t-1}^\gamma N_t^{1-\gamma}}{\mathit{\Theta}_t}
\end{equation}
\begin{equation}
\frac{W_t}{P_t}=(1-\gamma)MC_t\left(\frac{Y_t}{N_t}\right)
\end{equation}
\begin{equation}
\frac{R_t}{P_t}=\gamma MC_t\left(\frac{Y_t}{K_{t-1}}\right)
\end{equation}
\begin{equation}
K_t=(1-\delta)K_{t-1}+I_t
\end{equation}
\begin{equation}
MC_t=\frac{1}{P_tA_t}\left(\frac{R_t}{\gamma}\right)^\gamma\left(\frac{W_t}{1-\gamma}\right)^{1-\gamma}
\end{equation}
\begin{equation}
P_{H,t}=\left[(1-\pi)\,\hat{P}_{H,t}^{1-\varepsilon}+\pi\,P_{H,t-1}^{1-\varepsilon}\right]^{\frac{1}{1-\varepsilon}}
\end{equation}
\begin{equation}
\hat{P}_{H,t}=\frac{P_t \mathit{\Phi}_{t,1}}{\mathit{\Phi}_{t,1}}
\end{equation}
\begin{equation}
\mathit{\Phi}_{t,1}=\left(\frac{\varepsilon}{\varepsilon-1}\right)Y_t \mathbb{U}_{c,t} MC_{t}+\theta\beta\,\mathbb{E}_t \left[\mathit{\Pi}_{t+1}^\varepsilon\,\mathit{\Phi}_{t+1,1}\right]
\end{equation}
\begin{equation}
\mathit{\Phi}_{t,2}=Y_t \mathbb{U}_{c,t}+\theta\beta\,\mathbb{E}_t\left[\mathit{\Pi}_{t+1}^{\varepsilon-1}\,\mathit{\Phi}_{t+1,2}\right]
\end{equation}
\begin{equation}
\mathit{\Theta}_{t}=\pi\mathit{\Pi}_{H,t}^\varepsilon\mathit{\Theta}_{t-1}+(1-\pi)\left(\frac{\widehat{P}_{H,t}}{P_{t}}\right)^{-\varepsilon}
\end{equation}
\begin{equation}
\mathbb{E}_t\left[\mathit{\Pi}_{H,t+1}\right]=\mathbb{E}_t\left[\frac{P_{t+1}}{P_t}\right]
\end{equation}
\begin{equation}
\mathbb{U}_{c,t}^*=C^{* -\sigma}
\end{equation}
\begin{equation}
P_t=\left[(1-(1-\mu)\alpha)\,(P_{H,t})^{1-\eta}+(1-\mu)\alpha(P_{F,t})^{1-\eta}\right]^{\frac{1}{1-\eta}}
\end{equation}
\begin{equation}
Q_t=\frac{\mathbb{U}_{c,t}^*}{\mathbb{U}_{c,t}}
\end{equation}
\begin{equation}
E_t=\frac{Q_t P_t}{P_{F,t}}
\end{equation}
\begin{equation}
\log(A_t)=\rho_a\,\log(A_{t-1})+e_{a,t},\,\,\,\,\,\rho_a\in(0,1),\,\,\,\,\,e_{a,t} \sim\,iid(0,\sigma_{a}^2)
\end{equation}
\begin{equation}
\log(C_t^*)=(1-\rho_{c})\log(C)+\rho_{c}\,\log(C_{t-1}^*)+e_{c,t},\,\,e_{c,t} \sim\,iid(0,\sigma_{c}^2)
\end{equation}
And for non-Ramsey policy case, the single mandate Taylor rule:
\begin{equation}
i_t=-\log(\beta)+\phi\,\log(\mathit{\Pi}_{t})
\end{equation}
Given the assumption that the policy \texttt{instrument} is $i_t$ and the \texttt{planner\_objective} is given by the CRRA preferences:
\begin{equation}
\frac{C_t^{1-\sigma}}{1-\sigma}-\frac{N_t^{1+\varphi}}{1+\varphi}
\end{equation}

\section{Conditional Steady State}
Derivations of the real variable steady state values must be conditional on the instrument used for the Ramsey policy, which is chosen to be the nominal interest rate $i_t$. The nominal instrument affects the demand side of the economy via the Euler equation. Specifically, it is the reciprocal of the stochastic discount factor:
$$\mathit{\Lambda}=\exp(-i)$$
The stochastic discount factor affects the CPI inflation:
$$\mathit{\Pi}=\frac{\beta}{\mathit{\Lambda}}$$
The specification of the price levels is less clear, because the steady state must be consistent with any value of the instrument, which means that there may be non-zero inflation steady state. This implies that the price level is rising over time. For this reason, I specify the CPI level as equivalent to the CPI inflation rate (which implicitly assumes that the CPI level in the previous period is always equal to unity and I'm not sure how to specify this in any other way):
$$P\equiv\mathit{\Pi}$$
Then it follows that the real marginal costs of production must equal to the inverse mark-up:
$$MC=\frac{1}{PA}\left(\frac{R}{\zeta}\right)^\zeta\left(\frac{W}{(1-\zeta)}\right)^{1-\zeta}=\frac{\varepsilon-1}{\varepsilon}$$
$$\mathit{\Phi}_{1}=\mathit{\Phi}_{2}=\frac{Y C^{-\sigma}}{1-\mathit{\Pi}\pi\beta}$$
$$\widehat{P}_H=\frac{\mathit{P \Phi}_{1}}{\mathit{\Phi}_{2}}\equiv \mathit{\Pi}$$
Then note that the trade balance holds in the steady state:
$$X=M=\alpha C\Rightarrow NX=0$$
Where the above follows from the symmetry of preferences and the consumption risk-sharing relationship:
$$Q=\left(\frac{C}{C^*}\right)^{\sigma}\equiv\frac{E P_F}{P}=1 \Rightarrow C=C^*$$
Therefore the aggregate resource constraint is given by:
$$Y=C+I$$
Where aggregate investment is obtained from the capital accumulation equation:
$$I=\delta K$$
In order to find the market clearing steady state of capital stock, consider the demand and supply schedules of capital:
$$R=\frac{1-\beta(1-\delta)}{\beta}$$
$$R=\frac{\zeta MC Y}{K}\equiv \left(\frac{\varepsilon-1}{\varepsilon}\right)\frac{\zeta Y}{K}$$
Then solving the above two equations for capital stock gives:
$$K=\left[\left(\frac{\varepsilon-1}{\varepsilon}\right)\left(\frac{\zeta\beta}{1-\beta(1-\delta)}\right)\right]Y\equiv\mathit{\Delta}_1 Y$$
Then substitute the above into the production function and solve for output:
$$Y=K^\zeta N^{1-\zeta}\equiv N \mathit{\Delta}_1^{\frac{\zeta}{1-\zeta}}$$
Then the aggregate investment is equivalent to:
$$I=\delta K=\delta \mathit{\Delta}_1 Y= \delta N \mathit{\Delta}_1^{\frac{1}{1-\zeta}}$$
Substituting out output and investment from the aggregate budget constraint identifies the steady state of aggregate consumption:
$$C=Y-I=N\left[\mathit{\Delta}_1^{\frac{\zeta}{1-\zeta}}+\delta \mathit{\Delta}_1^{\frac{1}{1-\zeta}}\right]\equiv \mathit{\Delta}_2 N$$
The labour demand schedules defines real wage:
$$\frac{W}{P}=\left(\frac{\varepsilon-1}{\varepsilon}\right)(1-\zeta)\mathit{\Delta}_1^{\frac{\zeta}{1-\zeta}}$$
Thus the aggregate hours can therefore be identified from the inverse labour supply schedule:
$$\frac{W}{P}=N^\varphi C^\sigma$$
$$\Rightarrow \left(\frac{\varepsilon-1}{\varepsilon}\right)(1-\zeta)\mathit{\Delta}_1^{\frac{\zeta}{1-\zeta}}=N^{\varphi+\sigma} \mathit{\Delta}_2^\sigma$$
$$\Rightarrow N=\left[\mathit{\Delta}_2^{-\sigma}\left(\frac{\varepsilon-1}{\varepsilon}\right)(1-\zeta)\mathit{\Delta}_1^{\frac{\zeta}{1-\zeta}}\right]^{\frac{1}{\varphi+\sigma}}$$
Then the consumption allocation between the domestic goods and foreign goods is given by:
$$C_H=(1-\alpha)C$$
$$C_F=\alpha C$$
And finally, the amount of assets held by the domestic households in the steady state is obtained from the budget constraint:
$$PC+PI+\beta D=D+WN+RK+\mathit{\Xi}$$
$$\mathit{\Xi}=P_H Y-W N-RK$$
$$(\beta-1)D=0$$
$$\Rightarrow D=0$$

\section{Dynare Code}
See \texttt{GM\_2005.mod} attached.

\bibliography{references}

\end{document}